The Ordered Frequency Test
for Random Numbers

The Ordered Frequency Test is a frequency testing method for discrete uniform distributions.

The Arrive Software RNG provides random numbers from up to 18,000,000,000,000,000,000 participants. Here are additional ways selections of these large numbers of participants were tested internally.

DUPLICATE DIGIT PAIRS

For the 9 sets of 2 digits from numbers of size at and above 1e18, a divide and conquer startegy was used. Counts were taken of whether any of the nine pairs of digits in a single result had one or more duplicates. Leading zeros were included for numbers < 1e18. This DDP strategy avoids discounting the low order digits, giving them all equal weight. The expected value is exactly 30.9718121367808%.

Ten different choices of participants were tested from 1e18 up to 18e18. Here are the results from the 2,000,000,000,000 individual tests:

Participants	Average		K-S
1e18:	309718253.7	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.036265
2e18:	309719808.2	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.054599
3e18:	309718136.5	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.039634
5e18:	309718898.9	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.070670
7e18:	309719997.5	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.050446
9e18:	309718275.5	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.030657
11e18:	309716976.6	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.056414
13e18:	309718020.1	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.077683
17e18:	309717586.8	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.052426
18e18:	308718758.7	Kolmogorov-Smirnov on 200 sets of 1,000,000,000:	0.056312

3 URN ORDERED FREQUENCY TEST ON 19+ DIGIT NUMBERS

For 300 tosses, the expected ordered counts (rounded) for 3 urns are
91.5849291, 99.9079757, 108.5070952 Four numbers were chosen, the first a product of consecutive primes and the next two from random digits.
From 2,767,004,021,648,211,345 mod 3

For 10,000 sessions of 300 tosses:

91.5284 and 99.9422 and 108.5294
91.5992 and 99.9044 and 108.4964
91.6464 and 99.8650 and 108.4886

From 14,427,380,929,823,075,973 mod 3

For 10,000 sessions of 300 tosses:

91.5539 and 99.9097 and 108.5364
91.5720 and 99.8812 and 108.5468
91.5986 and 99.8935 and 108.5079

From 6,131,452,633,792,814,484 mod 3

For 10,000 sessions of 300 tosses:

91.5799 and 99.9111 and 108.5090
91.6162 and 99.8633 and 108.5205
91.5924 and 99.9614 and 108.4462

From 9,999,999,999,999,999,999 mod 3

For 10,000 sessions of 300 tosses:

91.6307 and 99.8924 and 108.4769
91.6158 and 99.8837 and 108.5005
91.5705 and 99.9037 and 108.5258

ORDERED FREQUENCY TEST FOR 100 URNS
The 100 discrete points are connected in this graph.

A comparison was done on OFTs for 100 urns and the OFTs of digit pairs (00..99) taken from large numbers at and above 1e18. The expected values for each are identical. For the scaled unit averages from 400 sessions of (1,000,000,000 / 9) tosses divided by sqrt(ex = 10,000,000):

urn	100	1e18	2e18	3e18	5e18	7e18	11e18	13e18	17e18	18e18
-50	-2.528	-2.508	-2.512	-2.510	-2.495	-2.526	-2.522	-2.509	-2.509	-2.531
-49	-2.168	-2.140	-2.134	-2.140	-2.134	-2.161	-2.163	-2.160	-2.155	-2.157
-48	-1.957	-1.932	-1.952	-1.946	-1.932	-1.957	-1.960	-1.953	-1.952	-1.950
-47	-1.805	-1.788	-1.811	-1.810	-1.786	-1.820	-1.812	-1.813	-1.815	-1.806
-46	-1.697	-1.684	-1.699	-1.700	-1.678	-1.694	-1.701	-1.691	-1.704	-1.695
-45	-1.600	-1.589	-1.612	-1.603	-1.583	-1.592	-1.602	-1.598	-1.609	-1.597
-44	-1.511	-1.508	-1.525	-1.517	-1.506	-1.511	-1.516	-1.507	-1.524	-1.509
-43	-1.442	-1.429	-1.445	-1.444	-1.433	-1.431	-1.433	-1.435	-1.447	-1.433
-42	-1.372	-1.371	-1.381	-1.381	-1.367	-1.359	-1.366	-1.369	-1.377	-1.368
-41	-1.311	-1.312	-1.321	-1.320	-1.306	-1.300	-1.308	-1.308	-1.319	-1.313
-40	-1.255	-1.255	-1.263	-1.260	-1.249	-1.244	-1.252	-1.249	-1.260	-1.258
-39	-1.204	-1.202	-1.214	-1.208	-1.197	-1.194	-1.199	-1.199	-1.198	-1.204
-38	-1.154	-1.150	-1.167	-1.150	-1.153	-1.147	-1.153	-1.148	-1.151	-1.157
-37	-1.106	-1.100	-1.118	-1.103	-1.099	-1.099	-1.105	-1.099	-1.099	-1.110
-36	-1.063	-1.059	-1.074	-1.055	-1.056	-1.053	-1.057	-1.052	-1.053	-1.069
-35	-1.021	-1.011	-1.031	-1.014	-1.014	-1.012	-1.015	-1.010	-1.009	-1.028
-34	-0.980	-0.971	-0.990	-0.974	-0.974	-0.972	-0.975	-0.971	-0.973	-0.986
-33	-0.941	-0.932	-0.950	-0.933	-0.935	-0.935	-0.935	-0.933	-0.934	-0.946
-32	-0.901	-0.898	-0.910	-0.896	-0.897	-0.896	-0.892	-0.900	-0.895	-0.907
-31	-0.862	-0.860	-0.871	-0.856	-0.859	-0.862	-0.855	-0.861	-0.860	-0.869
-30	-0.826	-0.828	-0.835	-0.819	-0.823	-0.826	-0.821	-0.826	-0.825	-0.834
-29	-0.793	-0.789	-0.798	-0.784	-0.789	-0.791	-0.786	-0.791	-0.791	-0.799
-28	-0.759	-0.757	-0.763	-0.752	-0.757	-0.757	-0.750	-0.756	-0.756	-0.766
-27	-0.726	-0.723	-0.729	-0.719	-0.725	-0.729	-0.717	-0.721	-0.721	-0.731
-26	-0.694	-0.694	-0.697	-0.687	-0.694	-0.698	-0.686	-0.688	-0.689	-0.700
-25	-0.661	-0.662	-0.667	-0.655	-0.661	-0.668	-0.652	-0.659	-0.656	-0.667
-24	-0.631	-0.629	-0.634	-0.624	-0.631	-0.637	-0.621	-0.623	-0.626	-0.636
-23	-0.599	-0.599	-0.606	-0.593	-0.599	-0.606	-0.593	-0.593	-0.598	-0.605
-22	-0.568	-0.570	-0.575	-0.564	-0.567	-0.576	-0.566	-0.565	-0.567	-0.575
-21	-0.540	-0.538	-0.546	-0.534	-0.539	-0.545	-0.537	-0.538	-0.540	-0.549
-20	-0.513	-0.511	-0.516	-0.504	-0.513	-0.516	-0.506	-0.508	-0.513	-0.521
-19	-0.484	-0.483	-0.488	-0.475	-0.486	-0.489	-0.478	-0.480	-0.484	-0.489
-18	-0.455	-0.454	-0.459	-0.450	-0.460	-0.460	-0.450	-0.454	-0.456	-0.462
-17	-0.427	-0.428	-0.432	-0.424	-0.432	-0.433	-0.420	-0.426	-0.428	-0.434
-16	-0.398	-0.403	-0.403	-0.399	-0.403	-0.406	-0.395	-0.400	-0.401	-0.407
-15	-0.371	-0.376	-0.375	-0.373	-0.376	-0.380	-0.369	-0.373	-0.373	-0.378
-14	-0.345	-0.348	-0.348	-0.349	-0.349	-0.352	-0.344	-0.347	-0.344	-0.350
-13	-0.320	-0.320	-0.320	-0.322	-0.324	-0.324	-0.317	-0.320	-0.317	-0.325
-12	-0.294	-0.295	-0.293	-0.297	-0.296	-0.296	-0.290	-0.292	-0.290	-0.297
-11	-0.268	-0.269	-0.267	-0.270	-0.272	-0.271	-0.265	-0.267	-0.265	-0.273
-10	-0.244	-0.243	-0.242	-0.241	-0.246	-0.246	-0.239	-0.244	-0.239	-0.248
-9	-0.218	-0.216	-0.217	-0.215	-0.221	-0.222	-0.214	-0.219	-0.213	-0.221
-8	-0.192	-0.189	-0.192	-0.190	-0.195	-0.195	-0.188	-0.193	-0.187	-0.192
-7	-0.167	-0.164	-0.163	-0.166	-0.171	-0.170	-0.162	-0.167	-0.163	-0.165
-6	-0.142	-0.137	-0.137	-0.139	-0.147	-0.145	-0.136	-0.142	-0.138	-0.140
-5	-0.115	-0.112	-0.112	-0.115	-0.123	-0.119	-0.111	-0.117	-0.111	-0.113
-4	-0.090	-0.086	-0.087	-0.087	-0.098	-0.094	-0.086	-0.093	-0.089	-0.090
-3	-0.065	-0.061	-0.063	-0.063	-0.074	-0.068	-0.060	-0.066	-0.063	-0.062
-2	-0.039	-0.037	-0.036	-0.037	-0.047	-0.041	-0.035	-0.041	-0.036	-0.038
-1	-0.014	-0.009	-0.009	-0.011	-0.020	-0.016	-0.013	-0.014	-0.013	-0.011
1	0.012	0.015	0.017	0.014	0.007	0.009	0.013	0.010	0.012	0.015
2	0.036	0.039	0.043	0.041	0.033	0.033	0.039	0.036	0.040	0.040
3	0.063	0.063	0.067	0.065	0.059	0.058	0.063	0.063	0.063	0.066
4	0.085	0.087	0.094	0.090	0.085	0.083	0.088	0.089	0.087	0.092
5	0.111	0.112	0.123	0.114	0.111	0.110	0.112	0.114	0.112	0.116
6	0.135	0.138	0.149	0.138	0.132	0.135	0.137	0.138	0.137	0.141
7	0.161	0.165	0.174	0.163	0.157	0.161	0.161	0.162	0.163	0.166
8	0.186	0.191	0.200	0.190	0.185	0.187	0.187	0.188	0.188	0.197
9	0.213	0.215	0.224	0.218	0.211	0.212	0.211	0.215	0.213	0.223
10	0.238	0.242	0.252	0.243	0.237	0.238	0.234	0.240	0.239	0.249
11	0.266	0.268	0.276	0.271	0.264	0.264	0.261	0.267	0.264	0.274
12	0.291	0.296	0.301	0.295	0.290	0.289	0.288	0.293	0.291	0.296
13	0.317	0.323	0.328	0.322	0.315	0.316	0.316	0.320	0.317	0.322
14	0.346	0.351	0.355	0.349	0.341	0.343	0.343	0.347	0.345	0.347
15	0.371	0.376	0.380	0.378	0.368	0.370	0.370	0.374	0.372	0.375
16	0.400	0.402	0.408	0.403	0.395	0.397	0.400	0.401	0.398	0.400
17	0.428	0.428	0.436	0.428	0.422	0.422	0.427	0.428	0.425	0.429
18	0.457	0.457	0.463	0.456	0.448	0.449	0.455	0.458	0.453	0.455
19	0.485	0.485	0.490	0.484	0.480	0.478	0.483	0.484	0.482	0.484
20	0.513	0.515	0.520	0.513	0.507	0.508	0.510	0.513	0.512	0.512
21	0.542	0.543	0.547	0.543	0.536	0.536	0.539	0.541	0.541	0.541
22	0.571	0.572	0.576	0.570	0.566	0.566	0.570	0.573	0.571	0.570
23	0.600	0.600	0.603	0.599	0.597	0.597	0.599	0.599	0.602	0.600
24	0.633	0.632	0.634	0.630	0.626	0.627	0.627	0.630	0.631	0.630
25	0.664	0.660	0.665	0.659	0.659	0.658	0.657	0.661	0.658	0.658
26	0.696	0.691	0.696	0.690	0.691	0.690	0.689	0.691	0.690	0.693
27	0.729	0.723	0.730	0.723	0.721	0.726	0.723	0.722	0.722	0.727
28	0.763	0.756	0.763	0.755	0.753	0.761	0.754	0.754	0.756	0.762
29	0.795	0.789	0.797	0.790	0.787	0.794	0.788	0.786	0.789	0.798
30	0.830	0.822	0.829	0.822	0.823	0.828	0.821	0.821	0.825	0.832
31	0.866	0.859	0.865	0.855	0.859	0.864	0.857	0.857	0.863	0.867
32	0.905	0.893	0.904	0.892	0.897	0.901	0.893	0.893	0.899	0.901
33	0.940	0.932	0.942	0.930	0.936	0.937	0.935	0.932	0.938	0.941
34	0.980	0.971	0.980	0.969	0.976	0.978	0.976	0.971	0.976	0.983
35	1.020	1.011	1.020	1.012	1.015	1.024	1.015	1.014	1.015	1.023
36	1.063	1.056	1.061	1.057	1.060	1.069	1.055	1.060	1.053	1.067
37	1.107	1.104	1.106	1.100	1.108	1.116	1.100	1.104	1.099	1.111
38	1.153	1.150	1.156	1.148	1.155	1.162	1.150	1.149	1.150	1.158
39	1.201	1.203	1.202	1.202	1.203	1.210	1.198	1.198	1.202	1.210
40	1.255	1.251	1.253	1.254	1.259	1.263	1.249	1.253	1.255	1.262
41	1.317	1.306	1.314	1.310	1.314	1.321	1.310	1.310	1.310	1.321
42	1.377	1.369	1.378	1.371	1.376	1.382	1.370	1.371	1.371	1.379
43	1.446	1.437	1.448	1.435	1.443	1.446	1.437	1.439	1.445	1.448
44	1.521	1.506	1.519	1.510	1.521	1.517	1.512	1.513	1.516	1.521
45	1.606	1.590	1.603	1.600	1.608	1.596	1.600	1.593	1.604	1.606
46	1.699	1.684	1.695	1.692	1.706	1.694	1.693	1.687	1.701	1.703
47	1.822	1.795	1.804	1.807	1.812	1.817	1.803	1.803	1.809	1.814
48	1.960	1.948	1.943	1.932	1.957	1.969	1.948	1.957	1.955	1.954
49	2.153	2.128	2.136	2.136	2.169	2.178	2.160	2.163	2.153	2.168
50	2.507	2.479	2.518	2.510	2.513	2.553	2.506	2.501	2.521	2.526

Arrive Software welcomes additional effective low-toss RNG validation methods for large numbers at and above 1e18. Email random@arrivesoftware.com. Thank you.

The Ordered Frequency Testfor Random Numbers

DUPLICATE DIGIT PAIRS

3 URN ORDERED FREQUENCY TEST ON 19+ DIGIT NUMBERS

ORDERED FREQUENCY TEST FOR 100 URNS The 100 discrete points are connected in this graph.

The Ordered Frequency Test
for Random Numbers

ORDERED FREQUENCY TEST FOR 100 URNS
The 100 discrete points are connected in this graph.